The production of J/psi mesons in proton-proton collisions at sqrt(s)=7 TeV is studied with the LHCb detector at the LHC. The differential cross-section for prompt J/psi production is measured as a function of the J/psi transverse momentum pT and rapidity y in the fiducial region 0<pT<14 GeV/c and 2.0<y<4.5. The differential cross-section and fraction of J/psi from b-hadron decays are also measured in the same pT and y ranges. The analysis is based on a data sample corresponding to an integrated luminosity of 5.2pb-1. The measured cross-sections integrated over the fiducial region are 10.52 +/- 0.04 +/- 1.40 +1.64/-2.20 mub for prompt J/psi production and 1.14 +/- 0.01 +/- 0.16 mub for J/psi from b-hadron decays, where the first uncertainty is statistical and the second systematic. The prompt J/psi production cross-section is obtained assuming no J/psi polarisation and the third error indicates the acceptance uncertainty due to this assumption.
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